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Math Help - substitution

  1. #1
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    substitution

    The substitution x=sinēθ allows ∫ √ x/(1-x) to become ∫ 2sinēθdθ

    Find ∫ √ x/(1-x) between 0 and 1

    thanks
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  2. #2
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    Hello, gracey!

    The substitution x\,=\,\sin^2\!\theta allows \int \sqrt{\frac{x}{1-x}}\,dx to become: . \int 2\sin^2\!\theta\,d\theta
    . .
    This is true!

    Find: . \int^1_0\sqrt{\frac{x}{1-x}}\,dx
    Make the substitution and convert the limits . . .


    We have: . \int^{\frac{\pi}{2}}_02\sin^2\!\theta\,d\theta \;=\;\int^{\frac{\pi}{2}}_02\left(\frac{1-\cos2\theta}{2}\right)\,d\theta \;=\;\int^{\frac{\pi}{2}}_0(1-\cos2\theta)\,d\theta

    . .  =\;\theta - \frac{1}{2}\sin2\theta\,\bigg]^{\frac{\pi}{2}}_0 \;=\;\left[\frac{\pi}{2} - \frac{1}{2}\sin\pi\right] - \left[0 - \frac{1}{2}\sin0\right] \;=\;\boxed{\frac{\pi}{2}}

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